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PREPRINT. — This  preprint  is  subject  to  correction  and  modification 
and  is  not  to  be  republished  as  a whole  or  in  part  pending  its  formal  release 
by  the  American  Society  for  Testing  Materials  through  its  Secretary-Treasurer. 
It  is  issued  primarily  to  stimulate  written  discussion,  which  may  be  trans- 
mitted to  the  Secretary-Treasurer  for  presentation  at  the  approaching  Eight- 
eenth Annual  Meeting,  June  22  to  26,  1915. 


THE  FAILURE  OF  MATERIALS  UNDER 
REPEATED  STRESS. 


SUMMARY. 


Nearly  all  our  ideas  of  the  action  of  materials  under  stress 
are  derived  from  considerations  of  static  action.  The  action 
of  materials  under  repeated  stress  differs  from  static  action  in 
important  respects. 

Under  stresses  slightly  lower  than  the  elastic  limit  as  ordi- 
narily determined  materials  fail  under  oft-repeated  stress,  and 
the  relation  of  stress  at  failure  to  number  of  repetitions  may  be 
expressed  by  an  exponential  formula,  as  shown  by  Basquin 
and  others. 

A proposed  explanation  of  this  formula  is  given,  based  on 
the  structural  damage  done  to  material  by  inelastic  action  during 
application  and  removal  of  load. 

For  still  lower  stresses  no  positive  evidence  of  a fixed  endur- 
ance limit  has  been  found  for  commercial  materials,  and  various 
methods  used  for  locating  such  a supposed  limit  give  varying 
results  from  the  same  test  data. 

A modification  of  the  exponential  formula  is  proposed  for 
low  stresses. 

Tentative  values  of  constants  for  the  proposed  formulas 
are  given  and  also  some  general  discussion  on  action  under 
repeated  stress. 


THE  FAILURE  OF  MATERIALS  UNDER 
REPEATED  STRESS. 


By  H.  F.  Moore  and  F.  B.  Seely. 

Behavior  of  Materials. 

Materials  under  Static  Stress  and  under  Repeated  Stress. — The 
behavior  of  materials  under  repeated  stress  shows  important 
variations  from  the  action  under  static  stress.  Nearly  all  the 
ideas  of  repeated  stress  have  been  developed  from  considera- 
tions of  static  loading.  One  very  common  idea  is  that  for  any 
given  material  there  is  a definite  elastic  limit  below  which  the 
behavior  of  the  material  is  perfectly  elastic.  Under  static  loads 
such  a conception  may  be  regarded  as  exact  without  involving 
serious  error,  though  careful  writers  on  the  mechanics  of  mate- 
rials have  for  a long  time  recognized  that  no  absolute  elastic 
limit  has  ever  been  fixed  for  any  material.1  In  structures 
under  static  load  local  stresses  of  considerable  magnitude — 
frequently  beyond  the  yield  point  of  the  material — exist  without 
producing  any  appreciable  effect  on  the  stability  or  the  deforma- 
tion of  the  structure  as  a whole,  and  such  stresses  are  frequently 
neglected  in  structures  subjected  to  static  load.  If,  however, 
the  load  on  a structural  part  or  a machine  member  is  repeated 
many  times  such  local  overstress  may  cause  a crack  to  start 
which,  spreading,  eventually  destroys  the  member;  or  inelastic 
action  too  small  to  be  detected  even  by  delicate  static  tests  of 
material  may  by  cumulative  action  cause  serious  damage  under 
oft-repeated  loading. 

An  illustration  of  the  difference  between  static  and  repeated 
loading  is  furnished  by  the  action  of  wire  ropes  bent  around 
sheave  wheels.  The  fiber  stress  due  to  bending  is  high  and 
frequently  causes  inelastic  action,  which,  however,  is  confined 
to  a small  portion  of  each  wire.  This  inelastic  action  is  very 
difficult  to  detect  by  means  of  static  tests,  but  as  the  wire  is 

1 Thurston,  “Text-Book  of  Materials  of  Construction,”  p.  348;  Burr,  “The  Elasticity 
and  Resistance  of  Materials  of  Engineering,”  p.  200;  Wawrziniok,  “Handbuch  des  Materials- 
prufungwesen,”  p.  9;  Unwin,  “The  Testing  of  Materials  of  Construction,”  p.  13. 

(2) 


Moore  and  Seely  on  Repeated  Stress. 


3 


repeatedly  bent  around  sheave  wheels  this  high  local  stress  starts 
cracks  which  eventually  cause  rupture  of  individual  wires.  In 
this  case,  as  in  many  others,  the  conception  of  perfect  elastic 
action,  allowable  for  static  loading,  must  be  discarded  for  re- 
peated loads. 

Materials  under  Repeated  High  Stress. — For  a range  of 
fiber  stress  extending  from  the  yield  point  of  the  material  (for 
brittle  materials  the  ultimate  strength)  down  to  a stress  slightly 
lower  than  the  elastic  limit,  as  determined  by  laboratory  tests 
of  the  usual  precision,  repeated  stress  will  cause  failure,  and 
there  seems  to  exist  a fairly  definite  relation  between  fiber 
stress  and  the  number  of  repetitions  necessary  to  cause  failure. 
This  relation  was  pointed  out  by  Basquin1  before  this  Society. 
It  may  be  expressed  by  the  formula: 

S = KNq 

in  which  S = intensity  of  fiber  stress  in  pounds  per  square  inch, 
iV  = the  number  of  repetitions  of  stress  to  cause  failure,  and  K 
and  q are  experimentally  determined  constants.  A similar 
relation  was  noted  later  by  Eden,  Rose  and  Cunningham2  and 
by  Upton  and  Lewis.3  Whether  for  still  lower  stresses  such  a 
law  holds,  or  whether  there  is  an  “ endurance  limit”  below 
which  failure  will  not  occur  under  any  number  of  repetitions  of 
stress,  however  many,  is  a question  which  will  be  discussed  later. 

Within  the  stress  limits  named  above,  if  material  is  subjected 
to  a cycle  of  stress  involving  application  and  removal  of  load, 
delicate  measurements  of  deformation  will  show  that  the  rela- 
tion between  stress  and  deformation  is  represented  not  by  a 
single  straight  line,  but  by  two  curved  lines,  one  for  application 
and  one  for  removal  of  load.  Even  if  such  deviation  cannot  be 
detected  after  a single  cycle  of  stress  it  has  been  shown  by 
Bairstow4  that  the  deviation  may  become  appreciable  after 
several  thousand  repetitions,  and  that  the  stress-deformation 
curve  for  a cycle  of  stress  after  a few  repetitions  becomes  a closed 

lMThe  Exponential  Law  of  Endurance  Tests,”  Proceedings,  Am.  Soc.  Test.  Mats.,  Vol.  X, 
p.  625  (1910). 

2 “The  Endurance  of  Metals,”  Proceedings,  Inst.  Mech.  Engrs.  (British),  Parts  3 and  4, 
p.  389  (1911). 

* “The  Fatigue  Failure  of  Metals,”  American  Machinist,  Oct.  17  and  24,  1912. 

4 “The  Elastic  Limits  of  Iron  and  Steel  under  Cyclical  Variations  of  Stress,”  Philosophical 
Transactions,  Royal  Soc.,  A Vol.  210,  p.  35  (1910). 


4 


Moore  and  Seely  on  Repeated  Stress. 


curve  with  a general  form  like  that  shown  in  Fig.  1 ( a ).  These 
curves  of  stress-cycles  resemble  the  “ hysteresis”  curves  of  mag- 
netic material;  the  area  enclosed  by  the  loop  represents  energy 
lost  during  a cycle  of  stress,  and  this  loss  of  energy  is  spoken  of 
as  “ mechanical  hysteresis.” 

If  mechanical  energy  is  dissipated  during  a cycle  of  stress 
it  would  seem  that  the  lost  energy  must  be  transformed  into 
heat,  that  there  must  be  some  form  of  internal  friction  in  the 
material,  that  wear  and  structural  damage  take  place,  and  that 


Fig.  1. — (a)  General  Form  of  Stress- Deformation  Curve  for  a Cycle  of  Stress. 

(6)  Approximate  Form  of  Curve. 

if  the  action  is  continued  long  enough  the  material  will  be 
ruptured.  It  would  seem  reasonable  to  consider  the  amount 
of  structural  damage  during  a cycle  of  stress  to  be  proportional 
to  the  energy  transformed  into  heat,  or,  in  other  words,  to  the 
area  of  the  mechanical  hysteresis  loop.  The  shape  of  this  loop 
remains  similar  to  the  shape  developed  under  early  cycles  of 
stress.  Referring  to  Figs.  1 ( a ) and  ( b ),  this  area  is  seen  to  be 
very  nearly  equal  to  that  of  two  triangles  placed  base  to  base, 
or  to  \(Si—S^)e.  If  S2  is  denoted  by  QS i ( Q is  negative  if  the 
cycle  involves  reversal  of  stress)  the  structural  damage  done 
during  an  early  cycle  of  stress  (A')  is  given  by  the  equation: 

A'  =ASi(l—Q)e 


Moore  and  Seely  on  Repeated  Stress. 


5 


in  which  A = some  constant  and  e = the  width  of  the  hysteresis 
loop.  An  examination  of  the  various  tests  which  have  come 
under  the  writers’  notice,  especially  those  of  Bairstow,1  indicate 
that  e is  some  function  of  the  range  of  stress  (l—  Q)Si  and  may 
be  taken  as  proportional  to  5^(1  —Q)p  in  which  p is  some  con- 
stant. The  structural  damage  done  during  a cycle  of  stress  is 
then  given  by  the  equation: 

A‘=A'Siil+p)(l (1) 

in  which  A'  is  some  constant  to  be  determined  experimentally. 

The  damage  done  by  successive  cycles  is  not  the  same.  An 
examination  of  Bairstow’s  results,  of  the  results  of  tests  by 
Sondericker2  and  of  tests  in  the  Laboratory  of  Applied  Mechanics 


Fig.  2. — Relation  between  Damage  per  Cycle 
and  Number  of  Repetitions  of  Stress. 


of  the  University  of  Illinois  indicates  that  the  damage  A changes 
slightly  with  successive  cycles  and  that  the  relation  between 
damage  per  cycle,  A,  and  the  number  of  repetitions  of  stress, 
N , may  be  represented  by  the  equation: 

A = A'  Nn 

in  which  A'  = the  damage  done  per  cycle  during  the  earlier 
cycles  of  stress,  and  n = a constant  which  has  a very  small  numer- 
ical value.  This  equation  is  represented  by  a curve  similar  to 
that  shown  in  Fig.  2. 

1 “The  Elastic  Limits  of  Iron  and  Steels  under  Cyclical  Variations  of  Stress,”  Philosophical 
Transactions,  Royal  Soc.,  A,  Vol.  210,  p.  35  (1910). 

2 “Repeated  Stresses,”  with  a discussion  by  J.  E.  Howard,  Technology  Quarterly,  March, 
1899. 


6 


Moore  and  Seely  on  Repeated  Stress. 


The  total  structural  damage  done  by  any  number  of  repe- 
titions, N,  may  be  denoted  by  T,  and  will  be  represented  by 
the  area  under  the  curve  up  to  the  value  of  N or  by  the  expres- 
sion J~A  dN  which  is  equal  toJ~  A 'NndN  ( a is  some  small 

number — the  first  few  cycles  are  apt  to  be  irregular).  Integrat- 
ing the  above  expression  there  is  obtained : 

^ A' 


1 +n 


(iVr(1+w)  — a(1+w)) 


and  as  N is  large  as  compared  with  a this  may,  without  serious 
error,  be  written: 


T = 


A' 


2V(1+n) 


1 +U 

but  from  equation  (l) 

A'  =A'  Sj1+P\l 


hence 


T = 


0 (i+/o 

il/5i(1+#)(l-C)(1+^(1+*) 
1 +n 


If  the  structural  damage  spreads  regularly  across  the  most- 
stressed  cross-section  of  a member,  and  if  the  amount  of  struc- 
tural damage  necessary  to  cause  failure  be  denoted  by  W, 
then  at  failure  under  repeated  stress: 

_rr__  A rS^1  +p\  1 - Q) (1  +P)N{1  +n 1 
1 +n 


W = T = 


Grouping  the  experimental  constants  W,  A ',  1-j-p,  and  1 +n, 
denoting  W{lA-ri)/Af  by  B^l+P\  and  1+n/l+p  by  q,  solving 
for  Si  and  dropping  the  subscript,  there  is  obtained: 


(1  -Q)N* 

For  any  given  set  of  experiments  with  a constant  value  of 
Q,  the  stress  to  cause  failure  under  repeated  stress  might  be 
expected  to  be  proportional  to  some  negative  fractional  power 
of  the  number  of  repetitions,  which  result  is  in  harmony  with  the 
exponential  law  proposed  by  Basquin,  as  a result  of  his  direct 
study  of  results  of  tests  under  repeated  stress. 


Moore  and  Seely  on  Repeated  Stress, 


7 


Number  of  Repetitions  before  Rupture 


8 


Moore  and  Seely  on  Repeated  Stress. 


Equation  (2)  may  be  written: 
log  5 = log  B — log  (1  —Q)—q  lo gN (2a) 

Equation  (2)  is  represented  by  a straight  line  if  plotted  on 
logarithmic  cross-section  paper.  As  shown  in  Basquin’s  paper, 
also  by  Eden,  Rose  and  Cunningham,  and  by  Upton  and  Lewis, 
the  results  of  nearly  all  repeated  stress  tests,  if  plotted  on 
logarithmic  paper,  do  follow  a straight  line  up  to  a value  of  N 
slightly  greater  than  1,000,000.  In  Figs.  3,  4 and  5 are  given 
some  test  results  compiled  by  the  writers,  including  some  not 
given  in  Basquin’s  paper.  These  results  also  follow  a straight 
line  up  to  a value  of  N of  about  1,000,000;  the  corresponding 
value  of  S is  slightly  below  the  elastic  limit  which  would  be 
determined  by  a static  test  of  the  usual  laboratory  accuracy. 
The  writers  wish  to  acknowledge  the  loan  of  collected  test 
data  by  Professor  Basquin,  which  was  of  material  assistance. 

The  analytical  discussion  of  the  cumulative  damage  done 
by  repeated  stress,  which  has  been  given,  seems  to  yield  results 
in  accordance  with  the  results  of  tests,  and  is  submitted  as  an 
explanation  of  the  failure  of  materials  under  repeated  stress 
within  the  stress  limits  named;  that  is,  for  stresses  ranging  from 
the  yield  point  (or  ultimate  strength  for  brittle  materials)  down 
to  a stress  slightly  below  the  “ elastic  limit”  as  usually  deter- 
mined in  static  tests. 

Materials  under  Repeated  Low  Stress. — As  builders  of 
machines  have  to  design  parts  to  withstand  many  times  one 
million  repetitions  of  stress,  a problem  of  greatest  importance 
is  the  determination  of  the  action  of  repeated  stresses  lower 
than  those  considered  in  the  foregoing  paragraph.  Is  cumula- 
tive damage  done  under  these  lower  stresses,  and  will  they 
finally  cause  failure?  Or  is  there  an  “ endurance  limit”  below 
which  no  damage  is  done  to  the  material,  and  below  which  the 
material  will  withstand  an  infinite  number  of  repetitions? 

The  latter  view  is  the  one  which  has  been  widely  held,  and 
the  endurance  limit  has  been  regarded  as  coincident  with  the 
“true”  elastic  limit  of  a material.  In  favor  of  this  view  may 
be  cited  the  fact  that  the  number  of  repetitions  of  stress  neces- 
sary to  cause  failure  increases  very  rapidly  as  the  fiber  stress  is 
lowered;  that  a number  of  tests  have  been  made  in  which  test 


Moore  and  Seely  on  Repeated  Stress. 


9 


specimens  withstood  tens  of  millions  of  repetitions  of  stress 
without  failure;  and  that  at  low  stresses,  even  with  delicate 
measuring  apparatus,  there  can  be  detected  no  signs  of  struc- 
tural damage.  Various  methods  have  been  used  for  determining 


Fig.  5. — University  of  Illinois  Tests. 


the  value  of  the  endurance  limit  for  a material;  these  different 
methods  yield  widely  varying  results,  as  will  be  shown  in  detail 
later. 

In  favor  of  the  view  that  damage  is  done  to  materials  under 


10 


Moore  and  Seely  on  Repeated  Stress. 


low  stress  and  that  there  is  a probability  of  their  eventual  failure 
under  repeated  low  stress,  the  following  considerations  may  be 
cited: 

1.  The  occurrence  of  “slip  lines”1  in  metal  under  repeated 
stress  seems  to  be  the  result  of  cumulative  damage  within  a 
crystal  of  metal.  No  sharply  defined  lower  limit  has  been 
found,  either  for  the  appearance  of  these  slip  lines  or  for  their 
tendency  to  spread  and  develop  into  cracks. 

2.  The  gradual  development  of  permanent  set,  under 
repeated  stress  so  low  that  preliminary  static  tests  had  shown 
no  measurable  set,  seems  to  the  writers  to  be  an  indication  of 
damage  at  low  stresses.  This  development  is  shown  especially 
by  the  tests  of  Bairs  tow2  for  cycles  of  stress  not  involving  com- 
plete reversal.  Bairstow  found  that  the  set  for  any  stress 
gradually  increased,  though  for  a single  cycle  of  stress  no 
mechanical  hysteresis  could  be  detected;  and  that  after  several 
thousand  repetitions  the  set  did  not  further  increase  during 
the  test.  Whether  this  set  would  have  shown  further  increase 
with  an  increase  in  the  number  of  repetitions,  or  whether  increase 
would  have  been  shown  by  more  delicate  instruments,  is  an 
undecided  question.  In  the  opinion  of  the  writers  the  significant 
fact  is  the  cumulative  development  of  permanent  set  under 
repeated  low  stress. 

3.  The  sudden  sharp  breakages  which  occur  in  repeated 
stress  tests,  even  of  ductile  materials,  would  seem  to  indicate 
that  structural  damage  may  be  done  to  material  without  any 
undue  deformation  of  the  member  as  a whole.  The  fact  that  no 
undue  deformation  can  be  detected  is  no  sure  sign  that  a mate- 
rial is  free  from  danger  of  failure  under  repeated  stress. 

4.  Data  of  tests  involving  more  than  a million  repetitions 
of  stress  are  very  few,  yet  frequently  machine  parts  must  be 
designed  to  endure  several  hundred  millions  of  repetitions. 
The  repeated  stress  problems  of  the  time  of  Wohler  and  Baus- 
chinger  were  mainly  problems  of  railroad  bridge  members  and 
other  structures  and  machines  which  would  be  called  on  to  with- 

1 Ewing  and  Rosenhain,  “Crystallin  Structure  of  Metals,”  Philosophical  Transactions, 
Royal  Soc.,  Vol.  193,  p.  353  (1899);  Ewing  and  Humfrey,  “Effect  of  Strain  on  the  Crystallin 
Structure  of  Lead,”  Philosophical  Transactions,  Royal  Soc.,  Vol.  200,  p.  241  (1902). 

2 "The  Elastic  Limits  of  Iron  and  Steel  under  Cyclical  Variations  of  Stress,”  Philosophical 
Transactions,  Royal  Soc.,  A Vol.  210,  p.  35  (1910). 


Moore  and  Seely  on  Repeated  Stress. 


11 


stand  only  a few  million  repetitions  of  stress.  From  the  view- 
point of  these  earlier  investigators  experiments  under  a few 
million  repetitions  covered  the  ground;  for  machines  of  to-day 
reliance  on  the  results  of  such  experiments  involves  enormous 
extrapolation  of  test  results.  The  data  seem  hardly  sufficient 
for  establishing  an  endurance  limit  for  infinite  repetition,  or 
even  for  repetitions  numbering  hundreds  of  millions.  More- 
over, the  results  of  some  tests,  if  taken  alone,  seem  to  indicate 
that  some  exponential  law  of  endurance  holds  up  to  the  limit 
of  experimentation.  These  unusual  tests  are  discussed  later. 

5.  As  instruments  of  increased  delicacy  are  used  in  measur- 
ing deformation,  evidences  of  mechanical  hysteresis  are  found 
at  lower  and  lower  stresses  in  static  tests.1  In  actual  material 
these  evidences  have  been  found  at  stresses  not  much  above 
ordinary  working  stresses.  When  the  cumulative  action  of 
repeated  stress  is  considered,  the  indefiniteness  of  the  elastic 
limit  becomes  apparent.  While  the  statically  determined 
elastic  limit  has  some  significance  for  static  loading,  it  appar- 
ently has  no  significance  as  a criterion  of  endurance  strength. 

6.  If  elastic  vibrations  are  set  up  in  metal  test  specimens 
such  vibrations  soon  die  out.2  This  dying  out  would  seem  to 
indicate  loss  of  energy  in  heat,  with  accompanying  internal 
friction,  wear,  and  structural  damage. 

The  following  quotation  from  the  careful  work  of  Bair- 
stow3  is  of  interest: 

“For  equal  (reversed)  stresses  after  the  specimen  had  been  fixed  in 
position  and  before  it  had  been  loaded  in  either  direction  a reading  was  taken 
of  the  unstrained  length.  A similar  reading  was  recorded  after  the  tensile 
loads  had  been  applied  and  removed,  and  a third  reading  after  putting  on  and 
removing  the  compressive  load.  The  three  readings  were  alike  and  indicated 
complete  elasticity  within  the  accuracy  of  the  measurement.  The  stress, 
=*=  31,000  lb.  per  sq.  in.,  was  then  repeated  automatically,  and  for  some  time 
the  straight  line  continued  to  represent  the  cycle  of  extensions.  As  the 
number  of  repetitions  became  greater  the  ‘cyclical  permanent  set’  became 
measurable  and  gradually  increased  until  after  19,000  reversals  of  stress  it  had 

1 Moore,  “The  Physical  Significance  of  the  Elastic  Limit,”  Proceedings,  Internat.  Assn. 
Test.  Mats.,  Article  XXVIII  (1912). 

2 Boudouard,  “Break-Down  Tests  of  Materials,”  Proceedings,  Internat.  Assn.  Test. 
Mats.,  Article  V3?  (1912);  Kelvin,  “On  the  Elasticity  and  Viscosity  of  Metals,"  Proceedings, 
Royal  Soc.,  May  18,  1865;  also  “Collected  Mathematical  and  Physical  Papers,”  Vol.  III. 

* “The  Elastic  Limits  of  Iron  and  Steel  under  Cyclical  Variations  of  Stress,”  Philosophical 
Transactions,  Royal  Soc.,  A Vol.  210,  p.  35  (1910). 


12 


Moore  and  Seely  on  Repeated  Stress. 


become  1 1 per  cent  of  the  original  elastic  extension.  . . . Raising  the  stress 
to  33,000  lb.  per  sq.  in.  produced  an  immediate  increase  in  the  ‘cyclical  per- 
manent set’  . . . Finally  stresses  of  =*=  44,440  lb.  per  sq.  in.  were  imposed 
and  at  29,280  reversals  . . . the  width  of  the  hysteresis  loop  was  very  great, 
but  even  for  this  case  the  lines  (showing  release  of  loads,  both  tension  and 
compression)  are  parallel  to  the  original  elastic  line.  . . . 

“The  behavior  of  this  specimen  illustrates  the  necessity  for  Bauschinger’s 
hypothesis  relating  to  primitive  elastic  limits,  as  the  extensometer  was  incap- 
able of  showing  the  first  deviation  from  elasticity.  At  a slightly  lower  range, 
probably  =*=  28,600  lb.  per  sq.  in.,  the  specimen  would  have  been  really  elastic, 
as.no  number  of  reversals  would  have  produced  a hysteresis  loop.” 

The  writers  feel  that  the  conclusion  given  in  the  last  para- 
graph is  more  sweeping  than  is  justified  by  the  data,  and  that 
the  conclusion  in  such  a case  (which  is  typical)  should  rather 
have  been,  that  a slightly  lower  stress  applied  to  the  specimen 
would  have  required  many  more  repetitions  in  order  to  make 
it  evident  that  damage  was  being  done.  After  a study  of  the 
data  of  many  series  of  repeated  stress  tests,  the  writers  feel 
that  no  “ endurance  limit”  has  been  surely  found  below  which 
any  material  may  be  relied  on  for  indefinite  endurance.  It 
seems  to  the  writers  that  more  repetitions  of  stress  would  prob- 
ably have  broken  the  test  specimens,  more  delicate  instruments 
would  probably  have  shown  evidences  of  inelastic  action,  rise 
of  temperature,  or  other  sign  of  eventual  failure. 

To  the  writers  it  seems  that  a negative  argument  against 
the  use  of  a definite  endurance  limit  is  furnished  by  the  indefi- 
niteness of  its  determination.  A common  method  of  locating 
this  limit  is  to  plot  from  test  a curve  with  stresses  as  ordinates 
(. N ) and  number  of  repetitions  to  cause  failure  as  abscissas. 
This  method  is  shown  in  Fig.  6 (a)  for  a typical  set  of  test 
results  from  Wohler.  This  curve  becomes  nearly  horizontal  at 
a few  millions  of  repetitions  and  the  horizontal  line  to  which 
the  curve  is  asymptotic  is  judged  by  the  eye.  The  ordinate 
of  this  horizontal  line  is  taken  as  the  endurance  limit,  and  in 
this  case  gave  a value  of  18,000  lb.  per  sq.  in. 

Another  method  is  to  plot  stresses  as  ordinates  and  values 
of  1 /N  as  abscissas.  The  endurance  limit  is  taken  as  the 
ordinate  of  the  intersection  of  this  curve  (extended)  with  the 
zero  axis.  This  method  is  shown  in  Fig.  6 ( b ) and  for  the 
same  test  data  as  the  first  method  gives  a value  for  the  endur- 
ance limit  of  17,500  lb.  per  sq.  in. 


Stress  in  Thousand  Pounds  per  Square  Inch 


Moore  and  Seely  on  Repeated  Stress. 


13 


A third  method  is  to  plot  stresses  as  ordinates  and  some 
root  of  1/A  as  abscissas.  The  fourth  root  of  1/A  has  been  sug- 
gested by  C.  E.  Stromeyer.1  Fig.  6 (c)  shows  the  application 


30 
20 
to 
. o 


F 

j&2_ 

± 

0 20  40  60  80  100  120  *0 

Millions  or  Repetitions  op  St* ess  — N 
(a) 


30 

20 

10 

0 


30 

20 

10 

O 

30 

20 

/O 

0 


* 

£ 

* 

* 

j 

L 

2«/o7  4*i o7  , 6*io7  8* /o'7  i.2*/ob 

77  W 


Fig.  6. — Diagrams  Showing  Variation  in  Endurance  Limit  (. En ) with  Method 

of  Determination. 


of  this  method  to  the  given  test  data,  and  the  endurance  limit 
is  found  to  be  15,000  lb.  per  sq.  in.  Fig.  6 (d)  has  been  plotted 


1913. 


1 "Memorandum  of  the  Chief  Engineer  of  the  Manchester  Steam  Users  Association”  for 


14 


Moore  and  Seely  on  Repeated  Stress. 


with  values  of  the  eighth  root  of  1 /N  as  abscissas,  and  the  value 
of  the  endurance  limit  thus  found  is  7,000  lb.  per  sq.  in.  This 
series  of  tests  involved  one  test  at  19  million  repetitions  and 
one  at  132  millions.  If  these  various  methods  were  applied  to 
tests  covering  no  more  than  one  million  repetitions  of  stress  the 
results  would  show  still  greater  variation. 

Formulas. 

Modification  of  the  Exponential  Equation  for  Repeated  Stress. 
— The  exponential  equations  for  repeated  stress,  (2)  and  (2a), 
give  for  high  values  of  N,  stresses  which  are  lower  than  those 
which  material  has  withstood  in  several  tests.  This  was  shown 
by  Basquin,  by  Upton  and  Lewis,  and  is  illustrated  in  Figs. 
3,  4 and  5.  Some  modification  of  these  formulas  would  seem 
to  be  necessary  for  low  stresses  and  high  values  of  N.  If  a 
test  specimen  or  machine  member  fails  under  a few  repetitions 
of  high  stress  the  damage  done  by  each  cycle  of  stress  is  con- 
siderable; an  appreciable  proportion  of  the  cross-sectional  area 
is  affected  by  this  damage,  and  the  spread  of  damage  is  regular 
and  rapid.  If  a test  specimen  or  machine  member  is  sub- 
jected to  cycles  of  low  stress  the  damage  done  during  each 
cycle  is  much  less;  a much  smaller  proportion  of  the  area  is 
affected,  and  the  location  of  the  little  areas  affected  is  dependent 
on  the  homogeniety  of  the  material  and  the  regularity  of  dis- 
tribution of  stress.  For  example,  a short  cylinder  under  com- 
pression has  a more  uniform  distribution  of  stress  under  a load 
sufficiently  heavy  to  cause  firm  bearing  between  the  cylinder 
and  its  bed,  than  it  has  under  a load  so  light  that  the  extremely 
small  areas  projecting  from  the  end  of  the  cylinder  are  not  all 
mashed  flat.  Under  low  stress  the  rate  of  spread  of  damage 
becomes  somewhat  a matter  of  probability  and  chance.  The 
affected  areas  may  be  so  grouped  that  damage  proceeds  regularly, 
but  probably  they  will  be  somewhat  scattered;  the  damage  will 
proceed  with  corresponding  slowness  and  the  endurance  of  the 
member  will  be  increased.  The  writers  suggest  that  the  expo- 
nential equations,  (2)  and  (2a),  modified  by  the  addition  of  a 
“probability  factor,”  be  used  for  estimating  the  fiber  stress 
which  will  cause  failure  under  any  given  number  of  repetitions 
of  stress,  N,  and  range  of  stress,  Q.  They  suggest  as  such  a 


Moore  and  Seely  on  Repeated  Stress. 


15 


factor  1 -\-kNm  in  which  k and  m are  constants  to  be  derived 
from  experiment.  The  modified  exponential  formula  then 
becomes 


5 = 


B 


(1  +kNm) (3) 

or  log  S = \og  B— log  (l  —Q)  —q  log  A+log  (l  +kNm) (3a) 


These  formulas  seem  rather  formidable,  but  tabular  values  for 


Fig.  7. — Tests  of  Cold-Rolled  Steel.  Watertown.  Arsenal. 


1 -\-kNm,  and  graphical  charts  for  the  solution  of  the  formulas 
are  given  later  which  facilitate  their  use. 

The  variability  in  the  action  of  metals  under  repeated  stress 
is  well  illustrated  by  the  results  of  a series  of  tests  of  cold-rolled 
steel  made  at  the  Watertown  Arsenal.1  The  results  of  this 
series  of  tests  is  plotted  logarithmically  in  Fig.  7.  It  will  be 
seen  that  for  the  higher  values  of  stress  the  range  of  values  of 
N is  less  than  for  low  values  of  stress.  A straight  line  fairly 


Tests  of  Metals,”  1893,  pp.  161-165,  170,  171,  176-178. 


16 


Moore  and  Seely  on  Repeated  Stress. 


well  represents  the  results  for  minimum  values  of  N ; this  straight 
line  corresponds  to  an  exponential  law,  equation  (2),  and  for 
such  minimum  values  it  would  seem  that  the  damage  caused  by 
repeated  stress  proceeded  regularly.  For  the  other  test  speci- 
mens the  damaged  areas  were  more  scattered,  or  for  some 
other  reason  damage  proceeded  more  slowly,  and  this  effect 
became  more  marked  at  lower  stresses. 

The  objection  may  be  raised  to  the  formula  proposed  by 
the  writers  in  that  it  involves  failure  of  any  material  under 
any  stress,  however  small,  if  that  stress  is  repeated  a sufficiently 
great  number  of  times;  and  it  may  be  stated  that  there  is  no 
proof  that  such  failure  will  occur.  It  has  not  been  conclusively 
proved  that  very  low  repeated  stresses  will  ultimately  break 
materials,  but  on  the  other  hand,  no  endurance  limit  for  indefi- 
nitely repeated  stress  has  yet  been  positively  established  for 
any  material.  There  is  practically  no  direct  experimental  evi- 
dence concerning  the  behavior  of  material  under  more  than 
100  million  repetitions  of  stress;  there  is  very  little  experimental 
evidence  for  more  than  10  million  repetitions,  yet  some  machine 
parts  must  be  designed  to  withstand  10  billion  repetitions  of 
stress. 

The  writers  believe  that  a formula  which  assumes  that  the 
same  destructive  action  which  occurs  under  high  stresses  will 
continue  to  act  with  diminished  intensity  under  low  stresses 
has  some  indirect  evidence  in  its  favor,  and  is  a safer  guide  for 
the  designer  than  is  a fixed  endurance  limit  below  which  destruc- 
tive action  is  assumed  to  cease.  It  should  be  noted  that  even 
for  values  of  N as  great  as  10  billion,  the  modified  exponential 
formulas  proposed  by  the  writers  give  stresses  which,  though 
somewhat  lower  than  the  values  usually  given  for  the  endurance 
limit,  are  yet  of  very  considerable  magnitude.  This  will  be 
discussed  under  the  head  of  “ Constants  for  the  Formulas.” 

In  addition  to  the  above  considerations,  it  should  be  noted 
that  for  very  long  endurance  of  machines  or  structures  deteriora- 
tion due  to  other  causes  than  repeated  stress — wear,  corrosion, 
change  of  current  practice,  and  the  like — becomes  a criterion 
of  endurance. 

Constants  for  the  Formulas.— The  writers  have  studied  the 
available  data  of  repeated  stress  tests  and  submit  tentative 


Moore  and  Seely  on  Repeated  Stress. 


17 


values  for  the  constants  in  the  formulas  proposed,  namely, 
equations  (2),  (2a),  (3)  and  (3a). 

B has  been  determined  for  a number  of  materials  from  the 
data  of  repeated  stress  tests.  It  is  the  value  of  the  ordinate 
for  N = 1 (line  extended  backward)  and  Q = — 1 (stress  com- 
pletely reversed).  In  Table  I are  given  tentative  values  of  B 
for  some  common  metals.  The  value  of  Q (ratio  of  minimum 
stress  to  maximum  stress)  is  generally  known  for  any  structural 
part  or  machine  member.  If  the  stress  is  wholly  or  partially 
reversed  Q is  negative.  If  Q approaches  +1  in  value,  care 
should  be  taken  that  the  safe  static  stress  is  not  exceeded; 
the  safe  static  stress  is  always  effective  as  a criterion  of 
safety. 


Table  I. — Values  of  the  Constant  B in  the  Exponential  Formulas 
for  the  Endurance  of  Materials  under  Repeated  Stress. 


Material. 

B 

Log  B 

Structural  Steel  and  Soft  Machinery  Steel 

110  000 

5.0414 

Cold-rolled  Steel  Shafting 

275  000 

5.4393 

Steel,  0.45  per  cent  Carbon 

175  000 

5.2430 

Wrought  Iron 

100  000 

5.0000 

Hard  Steel 

250  000 

5.3979 

Hard-Steel  Wire 

400  000 

5.6021 

The  effect  of  range  of  stress  has  been  studied  in  connection 
with  the  work  of  Bauschinger  and  Wohler.  Equations  (2)  or 
(3)  gives  values  of  S for  complete  reversal  (Q—  — l)  one-half  as 
great  as  the  values  for  repetition  of  stress  from  zero  to  a maxi- 
mum (Q  = 0).  This  seems  to  agree  fairly  well  with  test  results, 
but  the  effect  of  range  of  stress  furnishes  a most  promising  field 
for  further  experimental  study. 

A value  of  q = J seems  to  fit  fairly  well  the  results  of  a wide 
range  of  tests  of  various-size  test  specimens  of  different  metals 
tested  on  a variety  of  machines.  Considerable  variation  from 
this  value  may  be  found,  but  the  writers  have  been  able  to  find 
no  systematic  variation;  and  as  was  pointed  out  by  Basquin1 

1 “The  Exponential  Law  of  Endurance  Tests,”  Proceedings,  Am.  Soc.  Test.  Mats.,  Vol.  X, 
p.  625  (1910). 


18 


Moore  and  Seely  on  Repeated  Stress. 


a considerable  variation  in  q makes  a comparatively  small 
variation  in  the  value  of  S for  a given  value  of  N. 

Values  of  constants  for  the  probability  factor  \+kNm 
must  be  very  largely  a matter  of  judgment.  Referring  again 
to  the  series  of  tests  shown  in  Fig.  7 the  minimum  values  of  S 
follow  an  exponential  relation  (shown  by  the  straight  line). 
This  relation  corresponds  to  a value  of  1 -\-kNm  of  unity  for 
all  values  of  N.  As  noted  previously,  for  values  of  N above 
one  million,  such  a result  seems  unusual  though  it  is  sometimes 
found.  As  our  test  data  for  long-time  endurance  tests  are  few, 
for  structural  parts  and  machine  members,  whose  failure  would 
endanger  life,  it  would  seem  advisable  to  use  a probability 
factor  of  unity  for  all  values  of  N in  which  case  the  proposed 
formula  for  repeated  stress  becomes: 


•(4) 

(4a) 


or  log  S = log  B — log(l  — Q)  — f log  N 


For  cases  in  which  failure  would  not  endanger  life,  the 
writers  propose  as  tentative  values,  0.015  for  k and  f for  m. 
These  values  were  chosen  after  a study  of  the  scanty  data  of 
long  endurance  tests,  and  represent  values  of  S rather  lower 
than  those  given  in  some  of  the  long-time  tests  studied.  The 
formula  for  repeated  stress  for  cases  in  which  failure  would  not 
endanger  life  then  becomes: 


S = - — r (l+0.015iV») 

(1  -Q)NlK 


(5) 


or  log  .S  = logZ?  — log  (l  — Q)  — i logA+log(l +0.015 ....  (5a) 

The  determination  of  the  value  of  1+0.015  N*  is  rather 
cumbersome  and  in  Table  II  are  given  values  of  this  factor 
for  various  values  of  N.  Charts  for  the  graphical  solution  of 
equations  (4)  and  (5)  are  given  in  Figs.  8 and  9.  Fig.  8 is  for 
the  solution  of  equation  (4),  and  Fig.  9 for  the  solution  of  equa- 
tion (5).  The  method  of  using  either  Fig.  8 or  Fig.  9 is  as  follows: 

Enter  the  diagram  at  the  bottom  at  the  given  value  of  Q 
for  the  problem,  pass  vertically  to  the  desired  value  of  N (this 
will  in  general  lie  between  two  curves,  and  its  location  must  be 


Moore  and  Seely  on  Repeated  Stress. 


19 


20 


Moore  and  Seely  on  Repeated  Stress. 


judged  by  interpolation).  From  this  point  pass  horizontally 
to  an  intersection  with  the  diagonal  corresponding  to  the  value 
of  B for  the  material,  then  from  this  intersection  vertically  to 
the  upper  edge  of  the  diagram,  where  the  value  of  S may  be 
read  off.  It  should  be  remembered  that  5 is  the  breaking  stress. 

Factor  of  Safety  for  Repeated  Stress. — In  design  involving 
static  conditions  the  working  stress  must  be  less  than  the 
ultimate  strength  of  the  material.  Reduction  of  stress  by 
means  of  the  so-called  factor  of  safety  is  the  only  means  through 
which  a safe  design  is  assured.  It  has  already  been  pointed 
out  that  in  no  case  must  the  law  of  fatigue  be  considered  to 


Table  II. — Values  of  the  Probability  Factor  (1+0.015  N*) 
for  Various  Values  of  N. 


N 

(1+0.015  JVs) 

N 

(1+0.015  Nh) 

100  000 

1.063 

100  000  000 

1.150 

500  000 

1.077 

500  000  000 

1.183 

1 000  000 

1.084 

1 000  000  000 

1.200 

5 000  000 

1.103 

5 000  000  000 

1.244 

10  000  000 

1.112 

10  000  000  000 

1.267 

50  000  000 

1.138 

50  000.000  000 

1.326 

Note. — For  structures  and  machines  whose  failure  would  endanger  life  it  is  suggested 
that  the  value  of  the  probability  factor  be  taken  as  unity  for  all  values  of  N.  For  other 
structures  or  machine  parts  the  values  as  given  by  the  above  expression  and  here  tabulated 
are  suggested. 


hold  for  stresses  greater  than  the  yield  point  of  the  material, 
and  that  the  requirements  of  static  design  must  always  be  met. 

When  the  endurance  strength  of  a machine  member  is 
less  than  the  yield  point  of  the  material,  a factor  of  safety  must 
be  applied  to  it  in  order  to  obtain  a safe  working  repeated  stress. 
Fatigue  involves  two  factors,  stress  and  the  number  of  repeti- 
tions, and  the  factor  of  safety  may  be  applied  either  to  the  stress 
or  to  the  number  of  repetitions.  Since  the  stress  must  also 
satisfy  the  requirements  for  proper  static  design,  the  “life”  of 
the  material  or  machine  would  seem  to  be  properly  insured  by 
applying  the  factor  of  safety  to  the  number  of  repetitions; 
that  is,  a design  should  be  made  with  a stress  corresponding  to 


Moore  and  Seely  on  Repeated  Stress. 


21 


failure  for  a number  of  repetitions  the  machine  is  to  withstand 
multiplied  by  x,  the  factor  of  safety.  This  places  emphasis 
upon  endurance  rather  than  on  strength,  but  since  the  stress 
is  low  when  determined  by  the  condition  of  fatigue,  this  emphasis 
seems  properly  placed.  While  this  method  reduces  the  stress 
considerably  less  than  would  the  application  of  the  same  factor 
of  safety  to  the  stress,  it  should  be  remembered  that  the  function 
of  the  factor  of  safety  is,  in  part,  to  guard  against  excessive 
stress  due  to  non-homogeneous  material,  initial  stresses,  and 
local  stresses  due  to  fabrication,  and  that  such  stresses  are 
reduced  in  effect  after  a few  repetitions.  Moreover,  excessive 
stresses  may  be  resisted  for  a considerable  length  of  time  with- 
out producing  more  than  a small  percentage  of  the  total  dam- 
age required  for  rupture. 

On  the  other  hand,  it  should  be  remembered  that  repeated 
stresses  tend  to  destroy  the  primitive  ductility  of  the  material. 
Material  has  failed  in  practice  under  extremely  low  repeated 
stresses,  suggesting  that  defective  material,  local  flaws,  and  the 
like  may  play  a much  greater  part  in  determining  the  endurance 
of  a material  than  they  do  in  determining  its  static  strength. 

Two  examples  of  the  use  of  the  proposed  formulas  are 
given : 

1.  The  eyebars  and  chords  of  a railroad  bridge  truss  are 
to  be  made  of  structural  steel,  and  should  be  designed  to  with- 
stand 2,000,000  repetitions  of  stress,  varying  from  dead  load 
to  dead  load  plus  live  load.  The  d.ead  load  is  about  one  quarter 
of  the  live  load.  What  value  of  working  stress  should  be  used? 

The  value  of  Q for  this  case  is  + 0.20,  the  value  of  B (taken 
from  Table  II)  is  110,000,  and  the  value  of  N may  be  taken 
as  six  times  2,000,000  or  12,000,000.  As  the  failure  of  the 
bridge  would  endanger  life  equations  (4)  or  (4a)  are  to  be  used. 
Substituting  the  above  values  in  equations  (4)  or  (4a),  there 
is  obtained  for  the  proper  value  of  5 = 17,900,  lb.  per  sq.  in. 
The  same  result  may  be  obtained  from  the  use  of  Fig.  8.  As 
the  safe  static  stress  would  hardly  be  taken  higher  than  16,000 
lb.  per  sq.  in.  it  is  seen  that  static  conditions  govern  the  design. 

2.  A line  shaft  is  to  be  designed  to  withstand  500,000,000 
complete  reversals  of  bending  stress  and  is  to  be  made  of  a 
special  steel  for  which  B has  been  experimentally  determined 


22 


Moore  and  Seely  on  Repeated  Stress. 


as  300,000.  What  fiber  stress  should  be  allowed?  The  yield- 
point  strength  of  the  material  is  90,000  lb.  per  sq.  in. 

As  the  failure  of  a line  shaft  does  not,  in  general,  involve 
danger  to  life,  use  equations  (5)  or  (5a)  or  Fig.  9.  Q=  — 1.0, 
B is  given  as  300,000:  Take  N as  six  times  the  designed  endur- 
ance or  as  three  billion.  Substituting  the  above  in  equation  (5a) 
or  using  Fig.  9 there  is  obtained  for  the  working  stress 
S = 12,100  lb.  per  sq.  in.  As  this  is  far  below  the  allowable 
static  stress  endurance  conditions  would  govern  in  this  case. 

Various  Effects. 

Effect  of  Rapidity  of  Repetition  of  Stress. — A certain  amount 
of  time  is  required  for  any  member  of  a machine  or  structure 
to  assume  the  deformation  corresponding  to  any  given  load, 
and  if  repetitions  of  load  follow  each  other  at  intervals  shorter 
than  this  time,  the  deformation  in  the  member,  the  stress  set 
up,  and  the  number  of  repetitions  it  will  withstand  may  be 
appreciably  affected.  A few  recent  British  tests1  of  material 
under  repeated  stress  seem  to  indicate  that  for  small  mem- 
bers there  is  no  appreciable  effect  produced  by  varying  the 
rapidity  of  repetition  of  stress  below  about  2000  repetitions  per 
minute.  Above  that  speed  very  little  test  data  are  available. 

Effect  of  Rest  on  Resistance  to  Repeated  Stress. — If  metal  is 
stressed  beyond  the  yield  point  so  that  plastic  action  is  set 
up,  its  strength  and  its  elastic  action  are  improved  under  sub- 
sequent stress,  if  the  material  is  allowed  to  rest.  Recent  experi- 
ments by  British  investigators2  seem  to  indicate  that,  for  steel 
and  iron  at  least,  the  effect  of  rest  on  the  resistance  to  repeated 
stress  is  negligible  for  unit  stresses  below  the  yield  point  of  the 
material. 

Effect  of  Sudden  Change  of  Outline  of  Member. — Every 
sharp  corner  in  a piece  subjected  to  repeated  stress  facilitates 
the  formation  of  micro-flaws  in  the  piece.  From  results  of  re- 

1 Stanton  and  Bairstow,  “On  the  Resistance  of  Iron  and  Steel  to  Reversals  of  Stress,” 
Proceedings,  Inst.  Civil  Engrs.  (British),  Vol.  166,  p.  78  (1906);  also  Engineering  (London), 
Vol.  LXXIX,  p.  201  (1905).  Turner,  L.  P.,  “The  Strength  of  Steel  in  Compound  Stress  and 
Endurance  under  Repetitions  of  Stress,  Engineering  (London),  July  28,  August  11  and  25, 
Sept.  8,  1911. 

s Eden,  Rose  and  Cunningham,  “The  Endurance  of  Metals,”  Proceedings,  Inst.  Mech. 
Engrs.  (British),  Parts  3 and  4,  p.  389  (1911). 


Moore  and  Seely  on  Repeated  Stress. 


23 


peated  stress  tests  made  by  Stanton  and  Bairstow,  at  the  British 
National  Physical  Laboratory,  on  test  specimens  of  different 
shape,  the  superiority  of  the  test  specimens  in  which  sharp 
corners  are  avoided  is  obvious.  The  relative  values  for  strength 
under  repeated  stress  for  the  shapes  tested  seems  to  be  about 
as  follows: 


Rounded  fillet 100 

Standard  screw  thread  70 

Sharp  corner 50 


Service  Expected  from  Various  Machine  and  Structural 
Parts. — We  do  not  know  with  certainty  whether  any  material 
can  resist  an  infinite  number  of  repetitions  of  any  stress  however 
small.  The  safest  view  for  an  engineer  to  take  seems  to  be  that 
under  repeated  stress  materials  of  construction  have  a limited 
“life.”  The  exponential  formula  for  repeated  stress  gives  results 
in  accordance  with  this  view.  If  this  view  is  held,  the  number 
of  repetitions  which  any  structural  or  machine  member  will 
have  to  withstand  in  normal  service  becomes  of  importance. 
The  following  list  gives  the  numbers  of  repetitions  of  stress 
which  may  be  expected  to  be  applied  to  various  machine  and 
structural  members.  The  list  is  intended  to  be  suggestive 
rather  than  to  serve  as  an  exact  guide. 

The  members  of  a railway  bridge  carrying  100  trains  per 
day  for  a period  of  50  years  would  sustain  about  1,826,000 
repetitions  of  stress.  The  stress  would  vary  from  the  dead- 
load stress  to  a live-load  stress  averaging  somewhat  below  that 
caused  by  the  passage  of  the  heaviest  locomotives. 

A railroad  rail  over  which  250,000,000  tons  of  traffic  passes 
would  sustain  something  like  500,000  repetitions  of  locomotive 
wheel  loads,  the  stress  being  slightly  more  severe  than  a repe- 
tition from  zero  to  a maximum.  The  rail  would  have  to  stand, 
in  addition  to  the  locomotive  wheel  loads,  something  like 
15,000,000  repetitions  of  stress  caused  by  car  wheel  loads.  The 
stresses  set  up  by  car  wheel  loads  would  be  about  half  as  great 
as  the  stresses  set  up  by  the  locomotive  wheel  loads. 

A mine  hoisting  rope  bent  over  three  sheave  wheels  and 
operating  a hoist  100  times  a day,  in  a term  of  service  of  five 
years  would  sustain  550,000  repetitions  of  stress.  If  the  sheave 


24 


Moore  and  Seely  on  Repeated  Stress. 


wheels  are  so  placed  that  they  reverse  the  direction  of  the 
bending  of  the  rope  the  range  of  stress  would  be  nearly  a com- 
plete reversal;  if  bending  takes  place  in  one  direction  only  the 
range  of  stress  is  from  nearly  zero  to  a maximum. 

The  piston  rod  and  the  connecting  rod  of  a steam  engine 
running  at  300  r.p.m.  for  10  hours  per  day,  300  days  per  year 
for  10  years,  sustains  540,000,000  repetitions  of  stress,  and  the 
range  of  stress  involves  almost  complete  reversal. 

A band  saw  in  hard  service  for  two  months  sustains  about 
10,000,000  repetitions  of  stress  varying  from  nearly  zero  to  a 
maximum. 

A line  shaft  running  at  250  r.p.m.  for  10  hours  a day,  300 
days  per  year,  sustains  during  a service  of  20  years  900,000,000 
repetitions  of  bending  stress  due  to  force  transmitted  by 
belts,  gears,  and  driving  chains.  The  stress  is  almost  com- 
pletely reversed.  It  should  be  noted  that  for  the  line  shaft 
the  torsional  stress  is  not  repeated  nearly  so  often  as  is  the 
bending  stress. 

The  shaft  of  a steam  turbine  running  at  3,000  r.p.m.  for 
24  hours  per  day,  365  days  in  a year  during  10  years  service 
sustains  15,768,000,000  reversals  of  bending  stress  caused  by 
the  weight  of  rotating  parts  and  the  tangential  force  of  the 
inrushing  steam. 


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25 


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26 


Moore  and  Seely  on  Repeated  Stress. 


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